What I didn't understand about the demo, was the time and story gap between ME 2 and ME 3. Shepard is in the brig and seems to know Vega, but we don't have a clue to why. Plus Shephard is back on earth, not in the Normandy. From what the moderator posted, the demo is the way it starts, with no explanation about the time and story gap--big oversight on BioWare's part IMO.
Spoilers (below) that explain the gap between ME 2 and the ME 3 demo intro.
Turns out part of the story comes from the DLC
Arrival, which I didn't get, so I only know the gist of what happened; Shepard blew up a Mass Effect relay to stop some Reapers but also destroyed a lot of Batarians in the process. And there's an exclusive comic book that I never heard about but only came across by doing some searches.
DLC Arrival:
Arrival - Mass Effect Wiki - Mass Effect, Mass Effect 2, walkthroughs and more
Spoiler: Video of the last part of Arrival which explains in part why Shepard is back on Earth:
Mass Effect 2 - Arrival Ending - 720p HD - YouTube
Comic book (explains where Vegas enters):
Mass Effect: Conviction - Imgur
Also, there's a interesting suggestion by a BioWare moderator: do a default play through of ME 3
before you import one of your ME 2 characters. This way, when you play through with your characters, you'll see some of the subtle things that show up from your actions in ME and ME 2 that don't appear with a default playthrough.